Apolynomial has 5 turning points is multiplied by another polynomial that has 3 zeros. what is the minimum degree of the new polynomial

A turning point is when the gradient changes sign, namely a local maximum or minimum.
An nth degree polynomial can have up to n-1 turning points.

For example, a straight line (first degree polynomial) does not have a turning point, since 1-1=0.
A parabola is a second degree polynomial, therefore it has 2-1=1 turning point.

If a polynomial has 5 turning points, it has a minimum degree of 5+1=6.

When two polynomials are multiplied together, the resulting degree of the product is the sum of the degrees of the original polynomials.